PA003F — zero

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact expanded PA statement

forall n. 0 <= n

Structural proof guide

Generated structural guide

Zero is below every natural number.

This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.

The proof proceeds by equality transport (1).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA003G beta_prefix_sum_trace_exists PA003W beta_prefix_product_trace_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

4 script commands · 3 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

01Fix variables and assumptionsL1–1

Work with arbitrary variables or the premises of the current implication.

L1
intro n

02Construct an explicit witnessL2–2

Supply the displayed value, then prove that it has the required property.

L2
exists n

03Calculate and transport equalitiesL3–4

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

L3
rewrite PA3
L4
refl

Library-wide reading audit

Original exact command ledger · 4 lines